Mathematical and Numerical Studies on Meshless Methods for Exterior Unbounded Domain Problems

TL;DR

This paper introduces a modified meshless method for exterior unbounded domain problems that improves numerical accuracy and reduces matrix ill-conditioning by integrating the modified Trefftz method, supported by mathematical proofs and numerical experiments.

Contribution

The paper proposes a modified MFS with proper basis functions based on the MTM, providing mathematical analysis and demonstrating improved accuracy and condition numbers.

Findings

Reduced condition numbers with MTM and MMFS

Enhanced numerical accuracy of MMFS over conventional methods

Mathematical proofs of solvability and optimal parameters

Abstract

The method of fundamental solution (MFS) is an efficient meshless method for solving a boundary value problem in an exterior unbounded domain. The numerical solution obtained by the MFS is accurate, while the corresponding matrix equation is ill-conditioned. A modified MFS (MMFS) with the proper basis functions is proposed by the introduction of the modified Trefftz method (MTM). The concrete expressions of the corresponding condition numbers and the solvability by these methods are mathematically proven. Thereby, the optimal parameter minimizing the condition number is also mathematically given. Numerical experiments show that the condition numbers of the matrices corresponding to the MTM and the MMFS are reduced and that the numerical solution by the MMFS is more accurate than the one by the conventional method.